Coherent feedback from dissipation: the lasing mode volume of random lasers
aa r X i v : . [ c ond - m a t . d i s - nn ] M a r Coherent feedback from dissipation:the lasing mode volume of random lasers
Regine Frank ∗ , Andreas Lubatsch , Johann Kroha Institut f¨ur Theoretische Festk¨orperphysik,KIT Karlsruhe, 76128 Karlsruhe, Germany ∗ email:[email protected] Physikalisches Institut, Universit¨at Bonn,Nussallee 12, 53115 Bonn, Germany
In any quantum or wave system dissipation leadsto decoherence. Therefore, it was surprising infirst instance when experiments on strongly lossyrandom lasers showed unambiguously by mea-surements of the photon statistics and of the las-ing mode volume that coherent feedback is pos-sible in such systems . In coherent-feedbacklasers the photons form a far-from-equilibriumcondensate in the sense that a single quantumstate is occupied by a macroscopic number ofphotons . We demonstrate that the lossy dy-namics of random lasers alone imply a finite lasingmode volume, thus resolving the puzzle about co-herent feedback without resonator. Our theory ofrandom lasing including nonlinear gain and gainsaturation predicts a characteristic dependence ofthis lasing mode volume on the pump intensity,which can be tested experimentally. The con-cept of a random laser (RL), i.e., lasing in a homoge-neously disordered, laseractive medium without cavity,was introduced early on by Letokhov. It was then fore-seen that light would be amplified by stimulated emissionwhile diffusing through the random medium and cover-ing an unlimited region in space. Consequently, a con-tinuum of all the modes whose frequency lies within thelaser transition line would participate in the lasing, lead-ing to an uncorrelated (Bose-Einstein) photon numberstatistics (distributed feedback) and homogeneous laseremission from the entire system volume. By contrast, itwas found in experiments on ZnO powder lasers that thelaser light emission occurs from spatially strongly con-fined lasing spots with sharp, discrete emission lines andPoissonian photon number statistics, all three featuresbeing unambiguous signatures of coherent feedback, i.e.of laser emission from a single, spatially confined mode .This spatial confinement is at the heart of understandingcoherent feedback and the formation of a non-equilibriumphoton condensate state in homogeneous random lasers.However, its origin remained obscure ever since its discov-ery, because the commonly known, possible confinementmechanisms cannot explain the observed experimentalfacts. Anderson localization (AL) of light in disorderedmedia can be ruled out, because the photonic scatter-ing mean free path ℓ is much longer than the wavelengthof light λ , so that AL does not occur in the systemsat hand, and the Ioffe Regel criterion is not met, i.e. kl >
1. Random microcavities, preformed accidentiallyin the disordered medium, are unlikely to be the origin of coherent feedback, because neither would their intrin-sic surface roughness allow for localized cavity modes toexist nor could they explain the observed dependence ofthe average lasing spot radius on the pump intensity .Here we present a genuinely new mechanism for gener-ation of a finite coherence or mode volume and coherentfeedback, based on dissipation at the system surface. Weshow that optical losses at the surface imply that thelight field in the bulk of the system can be coherent (cor-related) only over a finite length ξ . Hence, this defines avolume within coherent feedback occurs, leading to Pois-sonian photon statistics. We identify ξ with the averageradius of a lasing mode. It is dynamically generated bythe highly non-linear, lossy dynamics of open laser sys-tems and, as such, exists only in the lasing state. Aswe perceived, this confinement mechanism is closely re-lated to the causality of both the propagation of lightand the transport of light intensity. Moreover, this beinga dynamical effect, we predict ξ to have a characteristic,namely power law, dependence on the laser pump inten-sity, which is found to be in qualitative agreement withavailable experiments .As a RL system we consider specifically a layer ofcompressed powder of laser-active material, whereby thegrains act as random scatterers and at the same time asamplifying medium. The layer has a thickness d and ex-tends infinitely in the x-y plane, as depicted in Fig. 1a.The pump light covers a wide surface area and affects theentire layer, so that the pump intensity may be assumedhomogeneous across the entire volume of the consideredsystem. The laser material is characterized by an atomicfour-level scheme with transition rates γ ij as defined inFig. 1b, although any other laser type would be possible.In order to address the problem of the coherence vol-ume in random lasers it is essential to set up the equa-tion of motion for the coherent part of the radiation.This part is gerenerated by stimulated emission only, al-though the quantum dynamics of the gain medium in-cludes also spontaneous emission. The time evolutionof the electric radiation field can be parameterized byas E ( t, r ) = E ( t, r ) exp [ − iωt ], where ω/ π is the lightfrequency and E ( t, r ) exp [ − iωt ] an amplitude functionvarying slowly on the scale of 2 π/ω . We now observe thatrandom lasers have to be treated as open systems, i.e., theloss-induced damping time is always much shorter thanthe lifetime γ of the upper atomic laser level (class Blaser). In this strongly damped regime, the stimulatedpart of the polarization follows the electric field instan-taneously. Hence, the wave equation for the stimulatedemission is local in time and can be written in terms ofthe dielectric function ε ( r , E ( t, r )) as[ ε ( r , E ( t, r )) c ω + ∇ ] E ( t, r ) = 0 . Note that ε ( r , E ( t, r )) incorporates the full, nonlinearlaser dynamics through its dependence on the field am-plitude E ( t, r ) and depends on position r both explicitlybecause of the random position of the dielectric scatterersand implicitly through its dependence on E . The las-ing state is characterized by a negative imaginary partof ε ( r , E ( t, r )). It is a priori not known and thereforedetermined by solving a four level laser rate equationsystem (Siegmann) in steady state (see supplementaryinformations). ∂N ∂t = N τ P − N τ ∂N ∂t = N τ − (cid:18) τ + 1 τ nr (cid:19) N − ( N − N ) τ n ph ∂N ∂t = (cid:18) τ + 1 τ nr (cid:19) N + ( N − N ) τ n ph − N τ ∂N ∂t = N τ − N τ P N tot = N + N + N + N . The electronic transition rates from the ground stateto the uppermost state and from the lower lasing levelto the ground state are assumed to be large comparedto all other transitions. Thus we focus on the de-scription of the photon number n ph and the inversion n = N /N tot = γ P γ P + γ nr + γ ( n ph +1) according to theEinstein rate equations in steady state. γ P equals thepump rate, γ nr represents the nonradiative decays and γ = 1 /τ features the stimulated emission rate, whichis the inverse of the lifetime or relaxation time of theupper lasing level. The rsulting inversion of the occupa-tion number thus defines the microscopic optical gain andthe challenge is now to relate this gain to the coherent(correlated) part of the diffusing radiation. Therefor thelaser rate system is coupled to sophisticated model formicroscopic transport of light in disordered amplifyingrandom media ? –18 based on Vollhardt-W¨olfle theoryfor the transport of electrons , which allows for the con-sideration of the influences of the microstructure of theamlifying media. For the completeness of the descrip-tion of ramdom lasing it has been demanded earlier byFlorescu et. al. that the effects due to multiple scat-tering of light and the internal Mie-resonances duringthese scattering events must be considered. We includedthose contributions by generalizing the diagrammatic de-scription for the transport of light and we particularly fo-cussed on the influence of interferences. Therefore we hadto deal with a Dyson description of the propagation ofthe electromagnetic wave on the one hand and the Bethe-Salpeter-Equation for the propagation and dissipation ofthe energy density of light on the other. The completedescription can be found in the supplementary, but tointroduce the reader breefly to this fascinating subject,the Bethe-Salpeter-Equation for the two particle Green’sfunction or in other words the energy propagator ˆ Φ isgiven here - which looks a bit denaunting at first glancebut actually it is a treasure chest for the random lasertheory. ˆ Φ = (cid:16) ˆ G R ⊗ ˆ G A (cid:17) h ⊗ + ˆ γ ˆ Φ i . The Bethe-Salpeter-Equation is the approbiate equa-tion of motion for the coherent part of light intensityΦ. The Green’s function G itself describes the behav-ior of the elctromagnetic wave, whereas the so calledirreducible vertrex ˆ γ reveals all interactions and theselfconsistence-loop is closed by the selfdependency of ˆ Φ .It has to be stressed that the propagation of light inten-sity in diffusive or strong scattering regimes obeys thesecond law of thermodynamics and thus the time scalesof wave propagation and intensity propagation in factcan be separated. The results drawn from these consid-erations turned out to be really promising, because thediffusive processes, the interferences and localization pro-cesses can be clearly identified as separate contributionsin the thereby derived diffusion constant D (Ω) D (Ω) (cid:2) − i ΩRe ǫωτ a (cid:3) = D tot − τ a D (Ω) M ( ω ) . The first term on the right hand side D tot = D + D b + D s could be slightly overlooked on the search forinterference effects which are represented through M ( ω ),the so called memory kernel. A detailed study reveals,however, that the memory kernel cannot cause true local-ization, i.e. D = 0, in laser active media, and D tot cannotbe compensated. D reflects the bare diffusion, the dis-sipative renormalizations due to absorbtion and/gain inthe system are summarized in the term D b and D s . Af-ter all we remark that this diffusion constant for systems,which are in any way laser-active, will never vanish, andtherefore AL of light in such systems seems to be beyondthe question in the discussion of random lasing.From the microscopically derived correlation length ξ (see supplementary information) in steady state forstrong pumping (see Fig. 4) one can draw several sig-nificant conclusions for the characteristics of a randomlaser ξ = α √ P + ξ ∞ where α is a numerical constant an P the pumping. Itis found that the natural antagonists, the pump strength P and the loss at the surfaces play a severe though in firstinstance counterintuitive role. The coherence length ξ insteady state clearly shrinks with the increase of the pumpstrength P whereas dissipation or loss at the surfacesof the depicted slab geometry (see Fig. 1) in contraryincreases the correlation length. The meaning of bothresults become obvious when we go back to the roots ofthe idea of random lasing again. Dicke in 1968 proposedthe possibility of the photon bomb by means of an infiniteincrease of the k -modes in an amplifying media. Ourresults in contrast proove that the nature of the onset oflasing in amplifying random media is clearly described bya finite correlation length and thus a finite mode volumewhich marks the exponential decay of the spatially andspectrally coherent light intensity inside the ZnO powderslab (Fig 3b).Loss at the surface however leads to anincrease of the correlation length ξ . 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This work was supported by theDeutsche Forschungsgemeinschaft through FOR 557 andgrant no. KR1726/3-1,-2.
METHODS h ω γ P n n n n γ γ γ nr 21 γ FIG. 1: text a Random Laser setup. The pump diameter isassumed to be large compared to the measured emission. b Four level laser sceme. -0.5 -0.25 0 0.25 0.5
Z [ d ] D / D d = 40 r d = 80 r -0.5 -0.25 0 0.25 0.5 00.050.10.150.2 n FIG. 2: The full and self-consistent diffusion constant D inunits of the bare diffusion constant D is shown as a functionof the width Z across the sample for pump rate P/γ = 2.The sample width is set to be d = 40 r and d = 80 r respec-tively as indicated in the legend , where r is the radius of thescatterers. As discussed in the text, the ratio D/D remainspractically unchanged, since the relative change observed inthe graph is of the same order as the accuracy, involved in thenumerical evaluation. This implies that interference effects donot change as a function of the sample width.The diffusionconstant D can be seen to behave inversed compared to theinversion n . Both quantities show a weak dependency to thesample’s width. FIG. 3: a The correlation volume of the RL mode stronglydecays with encreasing pump strength. b Calculated photondensityFIG. 4: Correlation length ξ as a function of the inversesquare root of the pump rate, measured in units of the in-verse square root of the transition rate γ , at the samplesurface, i.e. Z = 0 . d .The blue lines serve as a guide to the eye, emphasizing the lin-ear behavior of the correlation length in this plot. This clearlyreveals an inverse square root behavior of ξξ